#### by Rodrigo Bañuelos and Burgess Davis

Don Burkholder had a long and distinguished career in mathematics. He
is recognized worldwide for his deep and lasting contributions to
martingale theory and its applications to other areas of mathematics.
As a person, colleague and friend, he was kind and generous, serving
as mentor and role model for many young people entering the field.
Burkholder published many of his foundational papers in IMS journals,
particularly *The Annals of Mathematical Statistics* and *The Annals of Probability*. His many contributions to probability and analysis have
been reviewed in three recent publications: (1)
“Don Burkholder’s
work on Banach spaces,” ~~(2) “Donald Burkholder’s
work in martingales and analysis,”~~ and (3)
“The
foundational inequalities of D. L. Burkholder and some of their
ramifications.” The first two articles, one written by
Gilles Pisier
and the other by
Rodrigo Bañuelos
and
Burgess Davis,
respectively,
appeared in *Selected Works of Donald L. Burkholder*, published
by Springer in 2010. The third article, written by Bañuelos,
appeared in the special volume *Don Burkholder: A collection of
articles in his honor*, of the *Illinois Journal of
Mathematics*, 2011. The reader is referred to these articles for an
in-depth look at Burkholder’s work, its applications and connections
to different fields in mathematics where his work has had, and
continues to have, an impact. Here, we briefly touch upon a few of his
papers.

In 1966, Burkholder published his celebrated
“Martingale transforms”
paper in *The Annals of Mathematical Statistics*. In the 1930s,
Marcinkiewicz
and
Paley
(in separate papers) proved an inequality for
the Haar system of functions in the unit interval, which is equivalent
to the boundedness of dyadic martingale transforms with the
predictable sequence taking values in __\( \{1, -1\}. \)__ Burkholder’s paper
extended this result to the general setting of martingales. This paper
was inspired by the seminal work of
Donald Austin,
published in the
same volume of *The Annals of Statistics*, which showed the
finiteness of the square function of __\( L^1 \)__ bounded martingales. The
groundbreaking paper,
“Extrapolation and interpolation of
quasilinear operators on martingales,” written with
Richard Gundy,
was published in *Acta Mathematica* in 1970. The paper
introduced the “good __\( \lambda \)__ method,” now important in many areas
of mathematics for comparing norms of operators, and used it to prove
the famous Burkholder–Gundy martingale inequalities which compare the
norms of the square and maximal functions for all continuous path
martingales and many other regular martingales. In a subsequent paper
Burkholder, together with Gundy and
Martin Silverstein,
used these
inequalities to solve a longstanding open problem of
Hardy
and
Littlewood
concerning conjugate functions. This revolutionary paper
inspired many harmonic analysts to learn probability and many
probabilists to learn harmonic analysis, greatly enriching both
fields.

For the next several years, Burkholder continued to work on
applications of martingale theory and produced many other influential
papers.
His 1977 publication in *Advances in Mathematics*, for
example, examines the exit times of Brownian motion from cones and
other domains in Euclidean spaces and applies this to the behavior of
harmonic functions. This paper influenced many subsequent papers on
this and related subjects.

In 1984 Burkholder returned to boundedness of martingale transforms.
In his *Annals of Probability* paper,
“Boundary
value problems and sharp inequalities for martingale transforms,” he
introduced the now called “Burkholder method.” This was a new
approach that reproved his 1966 boundedness of martingale transforms results with optimal constants and provided extensions to the
setting of Banach spaces. This allowed Burkholder and others to answer
many questions about the geometry of Banach spaces for which both
martingale transforms and Calderón–Zygmund singular integral
operators (most notably the Hilbert transform) acting on functions
taking values in those Banach spaces, are bounded on __\( L^p, \)__ __\( 1 < p <
\infty. \)__ The question of characterizing the Banach spaces for which
this property holds arose from work of
Bochner
and
Taylor
in the
1930s. The ideas introduced by Burkholder in this paper were far ahead
of their time and it took some 20 years for others to fully understand
and explore them. The ideas and techniques in the 1984 article have
been extensively used in recent years on problems in probability and
harmonic analysis where the tools of __\( L^2 \)__ theory are either not
available or lead to estimates which are not optimal. The wider
implications, in particular its application to sharp __\( L^p \)__ bounds for
singular integral operators and ramifications in quasiconformal
mappings, problems from the calculus of variations dealing with
rank-one convex and quasiconvex functions, problems on optimal control
and the theory of Bellman functions, are all topics of current
interest. One should note that these fields, on the surface, are far
removed from martingale theory.

Burkholder served as president of the IMS from 1975 to 1976 and was
editor of *The Annals of Mathematical Statistics* from 1964 to
1967. He served on multiple editorial boards and scientific advisory
committees for both the IMS and other scientific societies. He
supervised 19 Ph.D. students and several postdocs. He was elected
member of the National Academy of Sciences in 1992. He was a fellow of
the IMS, the American Academy of Arts and Sciences, the Society for
Industrial and Applied Mathematics, the American Association for the
Advancement of Science and an inaugural fellow of the American
Mathematical Society. In 1970 he was an invited speaker at the ICM
held in Nice, France.

Don Burkholder grew up on a farm in Nebraska during the years of the Great Depression and the Dust Bowl, which perhaps had something to do with his ferocious work ethic. After he retired from his distinguished professorship position at the University of Illinois he was asked if he was now taking it easy, to which he replied “I am, I no longer work on Sunday afternoons.” His character also showed in how kind and helpful he was to others and specially to young mathematicians. He was always encouraging and never intimidating.

Burkholder attended Earlham College where he met Jean, his wife and
the mother of their three children. He received his Ph.D. in
mathematical statistics from the University of North Carolina at
Chapel Hill in 1955 where he worked under the mathematical
statistician
Wassily Hoeffding,
one of the founders of nonparametric
statistics. His thesis, “On a certain class of stochastic
approximation processes,” published in *The Annals of Mathematical
Statistics*, extended work of
Julius Blum,
Kai Lai Chung,
Joseph Hodges,
and
Erich Leo Lehmann
on “asymptotic normality.” Immediately after
graduating he took a job at the University of Illinois at
Urbana-Champaign where the legendary
Joseph L. Doob,
working on
martingales and applications of probability to potential theory and
complex analysis, was a professor. Almost certainly influenced by
Doob, Burkholder soon turned his attention to these subjects.

During his entire career, Burkholder wrote only six joint research papers. His closest collaborator, and the only mathematician with whom he wrote more than one paper, was Richard Gundy. Their work had a deep and lasting impact in probability and its applications. Burkholder’s later work in the eighties on martingales taking values in Banach spaces led him to new techniques to prove sharp martingale inequalities. The latter have had applications in areas which on the surface are far removed from martingales. The wide use of Burkholder’s ideas, results and techniques, shows the universality of his mathematics, although he seemingly “only” touched martingales.